biring

There are two equivalent definitions for birings. Let $\mathsf{CRing}$ denote the category of commutative unital rings and ring homomorphisms.

**Definition 1.** A biring is a coring object in $(\mathsf{CRing}, \otimes)$.

**Definition 2.** A biring is a ring $A$ equipped with a lift of the functor $\mathsf{CRing}(A, -) \colon \mathsf{CRing} \to \mathsf{Set}$ through the forgetful functor to a functor $\mathsf{CRing} \to \mathsf{CRing}$.

Birings were introduced in this paper:

- D. Tall? and G. Wraith?, Representable functors and operations on rings,
*Proc. London Math. Soc.*3, pages 619–643, 1970. web

category: mathematical methods